Issue with Implementing ∂x + i ∂y in Spherical Coordinates on Sphere Surface

[MARTÍN IGNACIO RUBIO]

Hi,

I’m working on an equation for a complex field A on the surface of a sphere, and I need to transform the operator \partial_x + i \partial_y to spherical coordinates. To do this, I tried defining the spherical coordinate derivative to get a transformation from the cartesian coordinates (also, pondered with some trigonometric functions of phi and theta) (also weighted with trigonometric functions of phi and theta for a consistent transformation) as follows: 

dphi = lambda B: d3.Differentiate(B, coords_sph['phi']) 

dtheta = lambda B: d3.Differentiate(B, coords_sph['theta']) 

However, this approach raises the following error: NotImplementedError: No subclasses of <class 'dedalus.core.operators.Differentiate'> found for the supplied arguments: (<Field 5239994848>, <dedalus.core.coords.Coordinate object at 0x13862bd40>), {}

I also tried using components of a gradient as an alternative, with:

Comp1 = lambda B: d3.AngularComponent(d3.Gradient(B, coords))

Comp2 = lambda B: d3.AzimuthalComponent(d3.Gradient(B, coords))

But this resulted in: ValueError: Not supported

Here are my definitions for the basis and field A:

dealias = 3/2

stop_sim_time = 100

timestepper = d3.RK222

timestep = 2e-3  

dtype = np.complex128

# Bases and domain

coords = d3.S2Coordinates('phi', 'theta')

dist = d3.Distributor(coords, dtype=dtype)

basis = d3.SphereBasis(coords, (Nphi, Ntheta), radius=R, dealias=dealias, dtype=dtype)

# Fields

A = dist.Field(name='A', bases=basis)

I would really appreciate any guidance on how to correctly implement these derivatives in spherical coordinates.

Thanks,

Martin RS

[Galen Neil Wilcox]

Hi Martin,

I am no expert but you may have some luck taking dot products of unit vectors with the gradient, like etheta@grad(B) + i*ephi@grad(B). See this post for setting up the unit vectors.

Otherwise, I have been working with similar operators on the sphere, and it seems best to use the built-in options if possible. Maybe there's a way to express your operator as a combination of grad, cross, curl, etc.

Best,

Galen

[Keaton Burns]

Hi Martin,

Unfortunately forming the unit vector fields on the sphere will cause problems, since these are not smooth vector fields. Since e_x and e_y are smooth fields, you could actually form those (or really, their projections on to the tangent space of the sphere) and use dot products to compute dx and dy. But these would be coupled operators that you probably couldn’t use implicitly. As Galen says, its recommended to use the built-in vector operators, if possible.

Best,

-Keaton

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[MARTÍN IGNACIO RUBIO]

Hi Galen and Keaton,
Thanks a lot for your reply. It has been very helpful.

Best,

Martín

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